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Elliptic Limit Cycles of Two-Dimensional Autonomous Differential Systems

Elliptic Limit Cycles of Two-Dimensional Autonomous Differential Systems Differential Equations, Vol. 38, No. 10, 2002, pp. 1385–1392. Translated from Differentsial'nye Uravneniya, Vol. 38, No. 10, 2002, pp. 1303–1309. Original Russian Text Copyright c 2002 by Dolov, Pavlyuk. ORDINARY DIFFERENTIAL EQUATIONS Elliptic Limit Cycles of Two-Dimensional Autonomous Di erential Systems M. V. Dolov and Yu.V.Pavlyuk Nizhni Novgorod State University, Nizhni Novgorod, Russia Received May 15, 2001 Real systems of di erential equations dx=dt = P (x;y);dy=dt = Q(x;y)(1) admitting particular algebraic integrals, where P and Q are polynomials with max(degP; deg Q)= n  2; were considered by numerous authors. These studies go back to Darboux, Poincar e, and Erugin. As a rule, attention was mainly paid to cases in which n = 2 or 3 and system (1) has invariant algebraic curves of degree  2. It was shown in [1] that if n = 3, then a circle and an ellipse cannot simultaneously be limit cycles of a system of the form (1). This assertion cannot be generalized to the case n = 4. Indeed, consider the system dx=dt =(y )H;dy=dt = (y )H + bH; (2) y x where ;b 2 R, b 6=0, H = !  !  ! , and http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Elliptic Limit Cycles of Two-Dimensional Autonomous Differential Systems

Dolov, M.; Pavlyuk, Yu.
Differential Equations , Volume 38 (10) – Oct 10, 2004
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Publisher
Springer Journals
Copyright
Copyright © 2002 by MAIK “Nauka/Interperiodica”
Subject
Mathematics; Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1023/A:1022310410920
Publisher site
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Abstract

Differential Equations, Vol. 38, No. 10, 2002, pp. 1385–1392. Translated from Differentsial'nye Uravneniya, Vol. 38, No. 10, 2002, pp. 1303–1309. Original Russian Text Copyright c 2002 by Dolov, Pavlyuk. ORDINARY DIFFERENTIAL EQUATIONS Elliptic Limit Cycles of Two-Dimensional Autonomous Di erential Systems M. V. Dolov and Yu.V.Pavlyuk Nizhni Novgorod State University, Nizhni Novgorod, Russia Received May 15, 2001 Real systems of di erential equations dx=dt = P (x;y);dy=dt = Q(x;y)(1) admitting particular algebraic integrals, where P and Q are polynomials with max(degP; deg Q)= n  2; were considered by numerous authors. These studies go back to Darboux, Poincar e, and Erugin. As a rule, attention was mainly paid to cases in which n = 2 or 3 and system (1) has invariant algebraic curves of degree  2. It was shown in [1] that if n = 3, then a circle and an ellipse cannot simultaneously be limit cycles of a system of the form (1). This assertion cannot be generalized to the case n = 4. Indeed, consider the system dx=dt =(y )H;dy=dt = (y )H + bH; (2) y x where ;b 2 R, b 6=0, H = !  !  ! , and

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Differential Equations – Springer Journals

Published: Oct 10, 2004

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